Path-kipas Ramsey numbers

For two given graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the smallest positive integer $p$ such that for every graph $G$ on $p$ vertices the following holds: either $G$ contains $F$ as a subgraph or the complement of $G$ contains $H$ as a subgraph. In this paper, we study the Ramsey numbers $R(P_n,\hat{K}_m)$, where $P_n$ is a path on $n$ vertices and $\hat{K}_m$ is the graph obtained from the join of $K_1$ and $P_m$. We determine the exact values of $R(P_n,\hat{K}_m)$ for the following values of $n$ and $m$: $1\le n \le 5$ and $m\ge 3$; $n\ge 6$ and ($m$ is odd, $3\le m\le 2n-1$) or ($m$ is even, $4\le m \le n+1$); $6\le n\le 7$ and $m=2n-2$ or $m \ge 2n$; $n\ge 8$ and $m=2n-2$ or $m=2n$ or ($q\cdot n-2q+1 \le m\le q\cdot n-q+2$ with $3\le q\le n-5$) or $m\ge (n-3)^2$; odd $n\ge 9$ and ($q\cdot n-3q+1\le m\le q\cdot n-2q$ with $3\le q\le (n-3)/2$) or ($q\cdot n-q-n+4m\le q\cdot n-2q$ with $(n-1)/2\le q\le n-4).$ Moreover, we give lower bounds and upper bounds for $R(P_n ,\hat{K}_m)$ for the other values of $m$ and $n$

On path-quasar Ramsey numbers

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we first give an explicit formula for the path-star Ramsey numbers. Secondly, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1\vee F_m)$ for the cases $m\leq n$ and $m\geq 2n$, and for the case that $F_m$ has no odd component. Moreover, we give a lower bound and an upper bound for the case $n+1\leq m\leq 2n-1$ and $F_m$ has at least one odd component.Comment: 7 page

THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Generalized Ramsey numbers for graphs

This thesis contains new contributions to Ramsey theory, in particular results that establish exact values of graph Ramsey numbers that were unknown to date. Given two graphs F and H, the Ramsey number R(F,H) is the smallest integer N such that, for any graph G of order N, either G contains F as a subgraph, or the complement of G contains H as a subgraph.\ud \ud Burr showed back in 1984 that the problem of determining the exact value of R(F,H) for arbitrary graphs F and H is NP-hard. Using techniques and results from graph theory, since the early 1970s researchers have been trying to confirm the exact value of Ramsey numbers for many well-studied families of graphs, including cycles, wheels, stars, paths, trees, fans and kipases. This is the main motivation and approach behind most of the results in this thesis.\ud \ud The thesis contains nine chapters with new results (Chapters 2–10), together with an introductory chapter (Chapter 1). The first three of these chapters (Chapters 2, 3 and 4) deal with Ramsey numbers for cycles versus stars or wheels. The next four chapters involve Ramsey numbers for trees versus fans or wheels (Chapter 5), fans versus fans, wheels or complete graphs (Chapter 6), paths versus kipases (Chapter 7), and the union of some graphs (Chapter 8). We study planar Ramsey numbers in Chapter 9, star-critical and upper size Ramsey numbers in Chapter 10